Vinayak Vatsal

Professor

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Graduate Student Supervision

Doctoral Student Supervision

Dissertations completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest dissertations.

Modular symbols, Eisenstein series, and congruences (2013)

Let E and f be an Eisenstein series and a cusp form, respectively, of the same weight k ≥ 2 and of the same level N, both eigenfunctions of the Hecke operators, and both normalized so that a₁ = 1. The main result we seek is that when E and f are congruent mod a prime p (which may be a prime ideal lying over a rational prime p > 2), the algebraic parts of the special values L(E,χ,j) and L(f,χ,j) satisfy congruences mod the same prime. More explicitly, the congruence result states that, under certain conditions,τ(χ ̄)L(f,χ,j)/(2πi)^(j−1)Ω_f^(sgn(E)) ≡ τ(χ ̄)L(E,χ,j)/(2πi)^(j)Ω_E (mod p)where the sign of E is ±1 depending on E, and Ω_f^(sgn(E)) is the correspondingcanonical period for f. Also, χ is a primitive Dirichlet character of conductor m, τ(χ ̄) is a Gauss sum, and j is an integer with 0
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Special values of anticyclotomic L-functions (2013)

This thesis consists of four chapters and deals with two different problems which are both related to the broad topic of special values of anticyclotomic L-functions. In Chapter 3, we generalize some results of Vatsal on studying the special values of Rankin-Selberg L-functions in an anticyclotomic ℤ_p-extension. Let g be a cuspidal Hilbert modular form of parallel weight (2,...,2) and level N over a totally real field F, and let K/F be a totally imaginary quadratic extension of relative discriminant D. We study the l-adic valuation of the special values L(g,χ,½) as χ varies over the ring class characters of K of P-power conductor, for some fixed prime ideal P. We prove our results under the only assumption that the prime to P part of N is relatively prime to D. In Chapter 4, we compute a basis for the two-dimensional subspace S_(k/₂)(Γ₀(4N),F) of half-integral weight modular forms associated, via the Shimura correspondence, to a newform F ∈ S_(k₋₁)(Γ₀(N)), which satisfies L(F,½) ≠ 0. Here, we let k be a positive integer such that k ≡ 3 mod 4 and N be a positive square-free integer. This is accomplished by using a result of Waldspurger, which allows one to produce a basis for the forms that correspond to a given F via local considerations, once a form in the Kohnen space has been determined. The squares of the Fourier coefficients of these forms are known to be essentially proportional to the central critical values of the L-function of F twisted by some quadratic characters.

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Master's Student Supervision

Theses completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest theses.

Elliptic curves of prime conductor - an exploration of conjecture, data and bias (2023)

This thesis presents data collected from the Bennett-Gherga-Retchnizer dataset of elliptic curves of prime conductor over the rationals. For these curves, we analyze the behaviors of the invariants which appear in the Birch and Swinnerton-Dyer conjecture - we present comparisons to the literature and point out differences and similarities between the families of elliptic curves of prime and composite conductors. Moreover, we present heuristic and computational evidence towards biases in the ranks, conductors and coefficients of elliptic curves within the latter family.

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On some Diophantine equations and applications (2023)

The Lebesgue-Nagell equation, x² + D = yⁿ, n ≥ 3 integer, is a classical family of Diophantine equations that has been extensively studied for decades. Bugeaud, Mignotte, and Siksek [20] made a landmark contribution to its study by resolving the equation for all values of D satisfying 1 ≤ D ≤ 100. However, many cases where -100 ≤ D ≤ -1 remain an open problem. We build on the works of Carlos [6] and Chen [22] to establish new results in several cases from this regime. Our techniques involve a combination of linear forms in logarithms and the modular method applied to Q-curves.Bennett et al. [10] proved using the theory of Diophantine equations that the Fourier coefficients of the modular discriminant form, or equivalently, values of the Ramanujan tau function, are never equal to the power of an odd prime smaller than 100. We generalize these inadmissibility results to other Hecke newforms with rational integer Fourier coefficients and trivial mod 2 residual Galois representation. In doing so, we use some results about the Lebsgue-Nagell equation and several techniques, including the modular method, Thue-Mahler equations, and the Primitive Divisor theorem for Lucas sequences.

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